# Maximum Modulus Theorem

• Nov 15th 2011, 11:16 AM
tarheelborn
Maximum Modulus Theorem
Suppose that \$\displaystyle f\$ is entire and that \$\displaystyle |f(z)-1|<1\$ for all \$\displaystyle z\$ with \$\displaystyle |z|=1\$. Prove that \$\displaystyle f(z)\$ has no zeroes in the disk \$\displaystyle \Delta= \{z:|z|<1\} \$.
• Nov 15th 2011, 11:45 AM
FernandoRevilla
Re: Maximum Modulus Theorem
Quote:

Originally Posted by tarheelborn
Suppose that \$\displaystyle f\$ is entire and that \$\displaystyle |f(z)-1|<1\$ for all \$\displaystyle z\$ with \$\displaystyle |z|=1\$. Prove that \$\displaystyle f(z)\$ has no zeroes in the disk \$\displaystyle \Delta= \{z:|z|<1\} \$.

Hint Denote \$\displaystyle g(z)=f(z)-1\$ and apply the Rouche's Theorem to \$\displaystyle f(z)=g(z)+1\$ .
• Nov 15th 2011, 12:37 PM
tarheelborn
Re: Maximum Modulus Theorem
We haven't had Rouche's Theorem. I have the Maximum Modulus Theorem and I am not really sure how to apply that here.
• Nov 15th 2011, 12:50 PM
xxp9
Re: Maximum Modulus Theorem
The maximum modlus theorem states if g is holomorphic, |g| reaches its maximum only on the boundary.
Now g=f-1 is holomorphic, and |g|<1 on |z|=1, so |g|<1 in the disk |z|<1.
So |f|=|1+g|>=|1-|g||>|1-1|=0 in the disk.